Royal Navy ship HMS Bulwark has already helped save the lives of more than 4,000 migrants - and tonight it will rescue more.
There are reports another 3,000 migrants have left Libya today, many on boats which are unseaworthy and do not even have engines.
After being towed out to sea, they are left to drift for days by human traffickers.
Defence Secretary Michael Fallon, who is aboard the ship tonight, told ITV News Europe is still not doing enough to tackle the problem.
He said: "The whole of Europe really need to adopt a more comprehensive approach to get after the traffickers, to understand how they're making money out of this trade in human misery and to do more to discourage people from leaving these countries."
Political correspondent Carl Dinnen reports from HMS Bulwark:
Hundreds of migrants were rescued from the Mediterranean at the weekend - but their journey will not stop there, as they tell ITV News.
Britain's flagship HMS Bulwark has been involved in its "largest operation to date", plucking more than 1,000 migrants from makeshift boats
As HMS Bulwark continues to rescue migrants in the Mediterranean, David Cameron pledged that Britain would not shirk its responsibilities.